Mathematical models of synchronization and emergence

Includes bibliographical references (pages 59-60)In this thesis I discuss two widely used models which are called Cucker-Smale and Kuramoto.\ud The Cucker-Smale model is a prototype model for describing the emergence of\ud organization in the systems of interacting agents. It applies to phenomena such as birds flocking, fish schooling, and formation of groups in the systems of autonomous robots. The\ud Kuramoto model is to describe the synchronization process in the systems of coupled oscillators.\ud It applies to phenomena such as crickets chirping in unison. I will show that there\ud is a relationship between these two models in other words Kuramoto can be obtained as an\ud asymptotic limit of Cucker-Smale model. I also prove a theorem concerning the convergence\ud of solutions of the models with the same initial data. The paper will also demonstrate how\ud the models can be represented as gradient flows on manifolds of a certain energy functional